Topics from One-Dimensional Dynamics

Description

Book description One-dimensional dynamics owns many deep results and avenues of active mathematical research. Numerous inroads to this research exist for the advanced undergraduate or beginning graduate student. This book provides glimpses into one-dimensional dynamics with the hope that the results presented illuminate the beauty and excitement of the field. Much of this material is covered nowhere else in textbook format, some are mini new research topics in themselves, and novel connections are drawn with other research areas both inside and outside the text. The material presented here is not meant to be approached in a linear fashion. Readers are encouraged to pick and choose favourite topics. Anyone with an interest in dynamics, novice or expert alike, will find much of interest within. Reviews ‘… particularly useful for students/beginners in the field. Due to an extensive bibliography, it will also serve as a very good reference book.‘ Source: European Mathematical Society Newsletter 'This book is intended as a text for an advanced undergraduate or beginning graduate students. As well as providing a brief account on the fundamental concepts of analysis and dynamical systems (Chapters 1-4 and 7-8), and a thorough explanation of topological entropy for piece-wise monotone interval maps (Chapter 9, sometimes with original proofs), the book contains substantial parts on unimodal interval maps and one chapter on complex quadratic polynomials. The quality of this exposition is very good: the material is organized so that all proofs are of easily manageable length; motivational comments are helpful, and there are plenty of illustrative examples and exercises.' Source: Zentralblatt MATH Contents List of Figures Preface 1 Topological Roots 1.1 Basics from Topology 1.2 Middle Third Cantor Set 2 Measure Theoretic Roots 12 2.1 Basics of Lebesgue Measure on 1R 2.2 A Nonmeasurable Set 2.3 Lebesgue Measure of Cantor Sets 2.3.1 The Middle Third Cantor Set 2.3.2 Other Cantor Sets 2.4 Sets of Lebesgue Measure Zero 3 Beginning Symbolic and Topological Dynamics 3.1 Periodic Behavior 3.2 Nonwandering and W—Limit Sets 3.3 Topological Conjugacy 3.4 Transitive Behavior 3.5 Recurrence 3.6 Shift Spaces 4 Beginning Measurable Dynamics 4.1 Preliminaries 4.2 Measurable Maps on I 4.3 Poincare Recurrence 4.4 Ergodicity 4.4.1 Integration of Measurable Functions 4.4.2 Averaging Measurable Functions Along Orbits 4.4.3 A Connection to Topological Dynamics 5 A First Example: The 2°°Map 5.1 Logistic Family 5.2 A Bit of Combinatorics 5.3 Construction of the Cantor Set w(c,g) 5.4 Cantor Set and Adding Machines 5.5 A Toeplitz Sequence 6 Kneading Maps 6.1 Hofbauer Towers and Kneading Maps 6.2 First Uses of Kneading Maps 6.3 Shadowing 6.4 Examples of Kneading Maps 7 Some Number Theory 7.1 Farey Tree 7.2 Continued Fractions 7.3 Continued Fractions and the Farey Tree 8 Circle Maps 8.1 Circle Homeomorphisms 8.2 Degree One Circle Maps 8.3 Irrational Rotations and Return Maps 8.4 Cantor Thread 9 Topological Entropy 9.1 Basic Properties of Topological Entropy 9.2 Entropy of Subshifts 9.3 Lapnumbers and Markov Extensions 9.4 Lapnumbers and Entropy 9.5 Semiconjugacy to a Piecewise Linear Map 9.6 The Monotonicity Problem 10 Symmetric Tent Maps 10.1 Preliminary Combinatorics 10.2 w—Limit Sets 10.3 Phase Portrait 10.4 Measure Results 10.5 Slow Recurrence and the CE Condition 10.6 Attractors 10.7 Combinatorics and Renormalization 11 Unimodal Maps and Rigid Rotations 11.1 Adding Machines in Unimodal Maps 11.2 Rigid Rotations in Unimodal Maps— I 11.3 Rigid Rotations in Unimodal Maps — II 12 fi—Transformations, Unimodal Maps, and Circle Maps 12.1 fi—Transformations and fi—Expansions 12.2 Flip—Half—of—the-Graph Trick 12.3 A Relation Between Unimodal Maps and Circle Maps 12.4 Comparing fi-Transformations and Tent Maps 12.5 Ledrappier's Example 12.6 Maps with Slope + oo 14 Complex Quadratic Dynamics 14.1 Julia Sets and External Rays 14.2 The Mandelbrot Set 14.3 Itineraries and Hubbard Trees Bibliography Index